Equations Of A Circle Diameter 12 And Center On X-axis
To determine the equation of a circle, it's crucial to understand the standard form of a circle's equation. The standard form is given by (x - h)² + (y - k)² = r², where (h, k) represents the coordinates of the center of the circle, and r is the radius. This equation is derived from the Pythagorean theorem and represents all the points (x, y) that are a fixed distance (the radius) away from the center.
In the context of our problem, we are given that the circle has a diameter of 12 units. The diameter is the distance across the circle through its center, and the radius is half of the diameter. Therefore, the radius (r) of our circle is 12 / 2 = 6 units. This value of the radius is essential as it will be squared in the equation, giving us r² = 6² = 36. The right side of our circle's equation will thus be 36. This is a critical first step in narrowing down the possible answers and focusing on the correct form of the equation.
Another critical piece of information is that the center of the circle lies on the x-axis. Any point on the x-axis has a y-coordinate of 0. Therefore, the center of our circle will have coordinates in the form (h, 0), where h is some real number. This simplifies the standard equation of the circle, as the (y - k)² term becomes (y - 0)², which simplifies to y². This eliminates options that do not have a y² term or have a more complex y term. Understanding this constraint significantly reduces the complexity of the problem and helps in identifying the correct equation forms more efficiently. The x-coordinate, h, however, can vary, leading to different possible equations depending on the specific location of the center along the x-axis.
Now, let's carefully examine each equation provided in the question, keeping in mind the standard form of a circle's equation: (x - h)² + (y - k)² = r². We know that our circle has a radius of 6 units, so r² = 36, and the center lies on the x-axis, meaning the y-coordinate of the center is 0. This simplifies our equation to (x - h)² + y² = 36, where h is the x-coordinate of the center. We will use this information to assess each option.
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(x - 12)² + y² = 12 This equation represents a circle centered at (12, 0) because the standard form is (x - h)² + (y - k)² = r², where (h, k) is the center. The term (x - 12)² indicates that h is 12, and since there's no term being added or subtracted from y before squaring, k is 0. However, the right side of the equation is 12, which means r² = 12, implying a radius of √12. This contradicts our earlier finding that the radius should be 6 (r² = 36), making this equation incorrect. Thus, this option does not represent the circle described in the problem.
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(x - 6)² + y² = 36 This equation is in the standard form of a circle, (x - h)² + (y - k)² = r². Here, h is 6, k is 0, and r² is 36. This means the circle's center is at (6, 0), which lies on the x-axis as required. The radius is √36 = 6 units, which matches the given diameter of 12 units. Therefore, this equation accurately represents a circle with a diameter of 12 units and its center on the x-axis, making it a correct option. This is a crucial point to understand as it validates the application of the standard circle equation formula.
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x² + y² = 12 This equation can be rewritten as (x - 0)² + (y - 0)² = 12. This implies that the center of the circle is at the origin (0, 0), which lies on the x-axis. However, the right side of the equation, 12, represents r², so the radius is √12. Since the required radius is 6 (as the diameter is 12), this equation is incorrect because it does not match the necessary radius. The equation describes a circle centered at the origin but with a different size than what the problem specifies.
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x² + y² = 144 Similar to the previous equation, this can be written as (x - 0)² + (y - 0)² = 144. The center is at the origin (0, 0), which lies on the x-axis. The right side, 144, represents r², so the radius is √144 = 12. This radius is incorrect because the required radius is 6 (half of the diameter 12). This equation describes a circle centered at the origin, but with a radius twice as large as what the problem requires.
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(x + 6)² + y² = 36 This equation can be rewritten as (x - (-6))² + y² = 36. This is in the standard form (x - h)² + (y - k)² = r², where h is -6, k is 0, and r² is 36. Thus, the center of this circle is at (-6, 0), which lies on the x-axis. The radius is √36 = 6 units, matching the requirement. This equation accurately represents a circle with a diameter of 12 units and its center on the x-axis, making it a correct option. The key here is to recognize that (x + 6) is equivalent to (x - (-6)), allowing us to correctly identify the center's x-coordinate.
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(x + 12)² + y² = 144 This equation can be rewritten as (x - (-12))² + y² = 144. This is in the standard form of a circle's equation, where h is -12, k is 0, and r² is 144. The center of this circle is at (-12, 0), which lies on the x-axis. However, r² = 144 implies a radius of √144 = 12 units. This is incorrect because the required radius is 6 units. This equation represents a circle with the correct center location (on the x-axis) but with the wrong size.
After a thorough analysis of each option, considering the standard equation of a circle, the given diameter, and the location of the center on the x-axis, we can confidently identify the correct equations. The equations that accurately represent a circle with a diameter of 12 units and its center lying on the x-axis are:
- (x - 6)² + y² = 36
- (x + 6)² + y² = 36
These equations both satisfy the conditions outlined in the problem, demonstrating a clear understanding of how to apply the standard circle equation and interpret its parameters. Each correct equation corresponds to a circle of the specified size with its center positioned appropriately along the x-axis, highlighting the flexibility and precision of the standard circle equation in defining geometric figures.
